Question

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs $E_1=E_0\cos vt,E_2=E_{0_{}}\cos 1vt+120°,and{E}_3=E_0\cos 1vt-120°$over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of $450MW$of electric power, a realistic value.

a. What would be the current in each wire if the transmission voltage were $E_0=120V_{rms}$?

b. In fact, transformers are used to step the transmission-line voltage up to $500k{V}_{rms}$. What is the current in each wire?

c. Big transformers are expensive. Why does the electric company use step-up transformers?

Short answer

(a.) The value of current in each wire is $I_{rms}=1.25\times {10}^{6}A$.

(b.) When transformer are used then the current in each wire is $I_{rms}=300A$.

(c.) Copper may be a good conductor , but it still have some resistance.

Step-by-step solution

Part (a) Step 1: Given information

We are given that a commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs$E_1=E_0\cos vt,E_2=E_{0_{}}\cos 1vt+120°,and{E}_3=E_0\cos 1vt-120°$ over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of $450MW$of electric power, a realistic value.

What would be the current in each wire if the transmission voltage were $E_0=120V_{rms}$.

Part (a) Step 1: Explanation

We need to find what is the current in each wire when step the transmission line up to$500k{V}_{rms}$.

$$\begin{gathered} P=I_{rms}\times E_{rms}+{(I_{rms}\times E_{rms})}_2+{(I_{rms}\times E_{rms})}_{3}P=3I_{rms}\times E_{rms} \\ 450\times {10}^{-6}=3\times I_{rms}\times 120 \\ {I}_{rms}=1.25\times {10}^{6}A \end{gathered}$$

Part (b) Step 1: Given information

We are given that a transformers are used to step the transmission-line voltage up to $500k{V}_{rms}$We need to find What is the current in each wire.

Part (b) Step 2: Explanation

We use same equation

$P=I_{rms}\times E_{rms}+{(I_{rms}\times E_{rms})}_2+{(I_{rms}\times E_{rms})}_3$

$$\begin{gathered} P=3I_{rms}\times E_{rms} \\ 450\times {10}^{-6}=3\times I_{rms}\times 500\times {10}^3 \\ {I}_{rms}==300A \end{gathered}$$

Part (c) Step 1: Given information

We need to find Big transformers are expensive. Why does the electric company use step-up transformers?

Part (c) Step 2: Explanation

Because Copper may be a good conductor , but it still have some resistance .